Stochastic Integral

A stochastic integral is an integral whose integrand, integrator, or both are stochastic processes (Durrett 1996). It is commonly written . For a simple process taking the random variable on the interval , a basic definition is

The construction is then extended by a specified mode of convergence under hypotheses on and .

Different endpoint conventions or limiting procedures can produce different stochastic integrals. When is a Wiener process and uses information available at the left endpoint, the construction gives the Ito integral (Kendall 2005). Special definitions are needed because sample paths such as those of a Wiener process typically do not have bounded variation, so ordinary Stieltjes integration does not apply.